# Full text of "Treatise On Analysis Vol-Ii"

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```116       XIII    INTEGRATION

increasing sequence (gn) of functions belonging to J such that for each n
we have /„ £ gn and ju*(#M) g /**(/„) -f £. Then, if we put /= sup/n and

g = sup gn , we shall have geJ,f^g, n*(g) = sup ju*(^n) ^ sup /**(/„) + e

n                                                                                                               n

by virtue of (13.5.2), and finally fi*(f) ^ M*(#) ^ SUP /**(/«) + e- Since s was

R

arbitrary, this will complete the proof.

By definition, for each n there exists hneJ such that /„ <g An and
Ai*(/J ^ M*(*») ^ /**(/») + 2~"8. Put ^ = supC/fi, . . . 5 hn\ so that ^ e ^
(12.7.5). Clearly the sequence (gn) is increasing, and/n ^ gn . We shall show by
induction on n that

(13.5.7.1 )                    n*(ffn) ^ A**CA) + e(l - 2-").

This is evident from the definitions when n = 1. Suppose that (13.5.7.1) is
true, and remark that gn+1 = sup(hn+1, gn) and/,, ^ inf(/zn+1, ^rt). Since the
functions hn+1 and gn do not take the value — oo, we have

sup(ftn+ !, flfB) + inf(/in+ 1? 0n) = /zn+ 1 + ^

and therefore, by virtue of (13.5.3) and the fact that n*(0n) and ju*(/zrt+1) are
finite,

/**(**+ i)-

by the inductive hypothesis (1 3.5.7.1 ).                                              Q.E.D.

It should be noted that for a decreasing sequence (/„), it is not necessarily
true that ju*/mf/n\ = inf /**(/„), even if all the numbers n*(fn) are finite

(Section 13.8, Problem 13).

(13.5.8)   If(fn) is any sequence of functions ^0, then

(00        \             00

Z/.U EM*(/B).
n~l    /        n=l
```